For the function [tex]F(x)=\frac{1}{(x-8)^2}[/tex], (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes.
(a) Choose the correct graph below.
A.
B.
C.
D.
Answer (2)
The function F ( x ) = ( x − 8 ) 2 1 is a shifted version of x 2 1 with domain ( − ∞ , 8 ) ∪ ( 8 , ∞ ) and range ( 0 , ∞ ) . The vertical asymptote is at x = 8 and the horizontal asymptote is at y = 0 . The correct graph choice is graph C.
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The function F ( x ) = ( x − 8 ) 2 1 is a horizontal shift of the basic rational function f ( x ) = x 2 1 by 8 units to the right.
The graph of F ( x ) has a vertical asymptote at x = 8 and a horizontal asymptote at y = 0 .
The domain of F ( x ) is all real numbers except 8, or ( − ∞ , 8 ) ∪ ( 8 , ∞ ) , and the range is all positive real numbers, or ( 0 , ∞ ) .
The final answer is: Graph C, Domain: ( − ∞ , 8 ) ∪ ( 8 , ∞ ) , Range: ( 0 , ∞ ) , Vertical Asymptote: x = 8 , Horizontal Asymptote: y = 0 .
Explanation
Problem Analysis The function is F ( x ) = ( x − 8 ) 2 1 . We need to graph this function using transformations, determine its domain and range, and identify any vertical, horizontal, or oblique asymptotes.
Identifying Transformations The function F ( x ) is a transformation of the basic rational function f ( x ) = x 2 1 . Specifically, it's a horizontal shift of 8 units to the right. The basic function f ( x ) = x 2 1 has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0 .
Graphing the Function Applying the horizontal shift to the graph of f ( x ) shifts the vertical asymptote to x = 8 , while the horizontal asymptote remains at y = 0 . The graph will look like graph C.
Determining the Domain The domain of F ( x ) consists of all real numbers except where the denominator is zero. Thus, x − 8 = 0 , so x = 8 . The domain is all real numbers except 8, which can be written as ( − ∞ , 8 ) ∪ ( 8 , ∞ ) .
Determining the Range Since the square of any real number is non-negative, 0"> ( x − 8 ) 2 > 0 for all x = 8 . Therefore, 0"> F ( x ) = ( x − 8 ) 2 1 > 0 . The range is all positive real numbers, which can be written as ( 0 , ∞ ) .
Identifying Asymptotes The vertical asymptote is the line x = 8 . The horizontal asymptote is the line y = 0 . There are no oblique asymptotes since the degree of the numerator (0) is less than the degree of the denominator (2).
Final Answer (a) The correct graph is C. (b) The domain is ( − ∞ , 8 ) ∪ ( 8 , ∞ ) , and the range is ( 0 , ∞ ) .
(c) The vertical asymptote is x = 8 , and the horizontal asymptote is y = 0 . There are no oblique asymptotes.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how shifting a function affects its behavior can help model the trajectory of a projectile. If you have a function that describes the height of a ball thrown in the air, shifting the function horizontally can represent throwing the ball from a different location. Similarly, in economics, transformations can be used to model how changes in market conditions affect supply and demand curves. By understanding these transformations, you can make predictions about how the system will behave under different conditions. The function F ( x ) = ( x − 8 ) 2 1 is a simple rational function, and understanding its transformations can help in analyzing more complex rational functions.
